#8295648

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52753	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(17,18,24,28)(19,29,30,25)(20,27,21,26)(22,32,31,23)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(1,2,6,7)(3,5,9,14)(4,8,13,15)(10,12,11,16)(17,19,24,30)(18,25,28,29)(20,22,21,31)(23,26,32,27)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(5,14)(8,15)(12,16)(17,20,24,21)(18,26,28,27)(19,22,30,31)(23,25,32,29)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3,6,9)(2,5,7,14)(4,11,13,10)(8,16,15,12)(17,21,24,20)(18,27,28,26)(19,22,30,31)(23,25,32,29)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,15)(9,11)(14,16)(17,22)(18,23)(19,21)(20,30)(24,31)(25,26)(27,29)(28,32)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47);(1,5)(2,9)(3,7)(4,12)(6,14)(8,11)(10,15)(13,16)(17,23)(18,22)(19,26)(20,29)(21,25)(24,32)(27,30)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202844);